Question

Erica is participating in a road race. The first part of the race is on a 5.2-mile-long straight road oriented at an angle of 25∘ north of east. The road then turns due north for another 5.0 mi to the finish line.
In miles, what is the straight-line distance from the starting point to the end of the race?

Answer 1

Answer:

8.6 miles

Explanation:

In order to calculate the straight-line distance between the starting point and the ending point, we need to resolve each part of the displacement into two perpendicular directions, let's say horizontal (x) and vertical (y) directions.

In the first part, Erica moves for 5.2 miles 25∘ north of east. Resolving into the two directions:

$d_{1x} = 5.2 cos 25^{\circ}=4.7 mi\\d_{1y} = 5.2 sin 25^{\circ} =2.2 mi$

In the second part, Erica moves 5.0 miles north, so the components are

$d_{2x} = 0\\d_{2y} = 5.0 mi$

So the components of the net displacement are

$d_x = d_{1x}+d_{2x}=4.7+0 = 4.7 mi\\d_y = d_{1y}+d_{2y} =2.2+5.0 = 7.2 mi$

And the magnitude of the net displacement is

$d=\sqrt{d_x^2 +d_y^2}=\sqrt{4.7^2+7.2^2}=8.6 mi$